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Theorem lubfun 18349
Description: The LUB is a function. (Contributed by NM, 9-Sep-2018.)
Hypothesis
Ref Expression
lubfun.u π‘ˆ = (lubβ€˜πΎ)
Assertion
Ref Expression
lubfun Fun π‘ˆ

Proof of Theorem lubfun
Dummy variables π‘₯ 𝑠 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funmpt 6594 . . . 4 Fun (𝑠 ∈ 𝒫 (Baseβ€˜πΎ) ↦ (β„©π‘₯ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)π‘₯ ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)𝑧 β†’ π‘₯(leβ€˜πΎ)𝑧))))
2 funres 6598 . . . 4 (Fun (𝑠 ∈ 𝒫 (Baseβ€˜πΎ) ↦ (β„©π‘₯ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)π‘₯ ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)𝑧 β†’ π‘₯(leβ€˜πΎ)𝑧)))) β†’ Fun ((𝑠 ∈ 𝒫 (Baseβ€˜πΎ) ↦ (β„©π‘₯ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)π‘₯ ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)𝑧 β†’ π‘₯(leβ€˜πΎ)𝑧)))) β†Ύ {𝑠 ∣ βˆƒ!π‘₯ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)π‘₯ ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)𝑧 β†’ π‘₯(leβ€˜πΎ)𝑧))}))
31, 2ax-mp 5 . . 3 Fun ((𝑠 ∈ 𝒫 (Baseβ€˜πΎ) ↦ (β„©π‘₯ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)π‘₯ ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)𝑧 β†’ π‘₯(leβ€˜πΎ)𝑧)))) β†Ύ {𝑠 ∣ βˆƒ!π‘₯ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)π‘₯ ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)𝑧 β†’ π‘₯(leβ€˜πΎ)𝑧))})
4 eqid 2727 . . . . 5 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
5 eqid 2727 . . . . 5 (leβ€˜πΎ) = (leβ€˜πΎ)
6 lubfun.u . . . . 5 π‘ˆ = (lubβ€˜πΎ)
7 biid 260 . . . . 5 ((βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)π‘₯ ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)𝑧 β†’ π‘₯(leβ€˜πΎ)𝑧)) ↔ (βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)π‘₯ ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)𝑧 β†’ π‘₯(leβ€˜πΎ)𝑧)))
8 id 22 . . . . 5 (𝐾 ∈ V β†’ 𝐾 ∈ V)
94, 5, 6, 7, 8lubfval 18347 . . . 4 (𝐾 ∈ V β†’ π‘ˆ = ((𝑠 ∈ 𝒫 (Baseβ€˜πΎ) ↦ (β„©π‘₯ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)π‘₯ ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)𝑧 β†’ π‘₯(leβ€˜πΎ)𝑧)))) β†Ύ {𝑠 ∣ βˆƒ!π‘₯ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)π‘₯ ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)𝑧 β†’ π‘₯(leβ€˜πΎ)𝑧))}))
109funeqd 6578 . . 3 (𝐾 ∈ V β†’ (Fun π‘ˆ ↔ Fun ((𝑠 ∈ 𝒫 (Baseβ€˜πΎ) ↦ (β„©π‘₯ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)π‘₯ ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)𝑧 β†’ π‘₯(leβ€˜πΎ)𝑧)))) β†Ύ {𝑠 ∣ βˆƒ!π‘₯ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)π‘₯ ∧ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(βˆ€π‘¦ ∈ 𝑠 𝑦(leβ€˜πΎ)𝑧 β†’ π‘₯(leβ€˜πΎ)𝑧))})))
113, 10mpbiri 257 . 2 (𝐾 ∈ V β†’ Fun π‘ˆ)
12 fun0 6621 . . 3 Fun βˆ…
13 fvprc 6892 . . . . 5 (Β¬ 𝐾 ∈ V β†’ (lubβ€˜πΎ) = βˆ…)
146, 13eqtrid 2779 . . . 4 (Β¬ 𝐾 ∈ V β†’ π‘ˆ = βˆ…)
1514funeqd 6578 . . 3 (Β¬ 𝐾 ∈ V β†’ (Fun π‘ˆ ↔ Fun βˆ…))
1612, 15mpbiri 257 . 2 (Β¬ 𝐾 ∈ V β†’ Fun π‘ˆ)
1711, 16pm2.61i 182 1 Fun π‘ˆ
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  {cab 2704  βˆ€wral 3057  βˆƒ!wreu 3370  Vcvv 3471  βˆ…c0 4324  π’« cpw 4604   class class class wbr 5150   ↦ cmpt 5233   β†Ύ cres 5682  Fun wfun 6545  β€˜cfv 6551  β„©crio 7379  Basecbs 17185  lecple 17245  lubclub 18306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-riota 7380  df-lub 18343
This theorem is referenced by:  joinfval  18370  joinfval2  18371
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